DOI QR코드

DOI QR Code

A STABILIZED CHARACTERISTIC FINITE VOLUME METHOD FOR TRANSIENT NAVIER-STOKES EQUATIONS

  • Zhang, Tong (School of Mathematics & Information Science, Henan Polytechnic University)
  • 투고 : 2010.09.11
  • 심사 : 2010.11.16
  • 발행 : 2011.09.30

초록

In this work, a stabilized characteristic finite volume method for the time-dependent Navier-Stokes equations is investigated based on the lowest equal-order finite element pair. The temporal differentiation and advection term are dealt with by characteristic scheme. Stability of the numerical solution is derived under some regularity assumptions. Optimal error estimates of the velocity and pressure are obtained by using the relationship between the finite volume and finite element methods.

키워드

참고문헌

  1. A.Allievi, R.Bermejo, Finite element modified method of characteristics for the Navier- Stokes equations, Int. J. Numer. Meth. Fluids, 32 (2000) 439-464. https://doi.org/10.1002/(SICI)1097-0363(20000229)32:4<439::AID-FLD946>3.0.CO;2-Y
  2. R.Bank, D.Rose, Some error estimates for the Box method, SIAM J. Numer. Anal., 24 (1987) 777-787. https://doi.org/10.1137/0724050
  3. P.Bochev, C.Dohrmann, M.Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal. 44 (2006) 82-101. https://doi.org/10.1137/S0036142905444482
  4. K.Boukir, Y.Maday, B.MEtivet, E.Razafindrakoto, A high-order characteristic/fnite ele- ment method for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 25 (1997) 1421-1454. https://doi.org/10.1002/(SICI)1097-0363(19971230)25:12<1421::AID-FLD334>3.0.CO;2-A
  5. Z.Cai, S.McCormick, On the accuracy of the finite volume element method for diffusion equations on composite grids, SIAM J. Numer. Anal., 27 (1990) 635-655.
  6. Z.Cai, J.Mandel, S.McCormick, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal., 28 (1991) 392-402. https://doi.org/10.1137/0728022
  7. Z.Chen, Finite element methods and their applications, Spring-Verlag, Heidelberg, 2005.
  8. Z.Chen, R.Li, A.Zhou, A note on the optimal $L^2$-estimate of the finite volume element method, Adv. Comput. Math., 16 (2002) 291-303. https://doi.org/10.1023/A:1014577215948
  9. P.Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Com- pany, 1978.
  10. J.Douglas, T.Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982) 871-885. https://doi.org/10.1137/0719063
  11. R.Ewing, T.Lin, Y.Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J.Numer. Anal., 39 (2002) 1865-1888. https://doi.org/10.1137/S0036142900368873
  12. F.Gao, Y.Yuan, The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem, Comput. Math. Appl., 56 (2008) 71-81. https://doi.org/10.1016/j.camwa.2007.11.033
  13. V.Girault, P.A.Raviart, Finite element method for Navier-Stokes equations: theory and algorithms, Springer-Verlag, Berlin, Herdelberg, 1987.
  14. G.He, Y.He, The finite volume method based on stabilized finite element for the stationary Navier-Stokes problem, J. Comput. Appl. Math., 205 (2007) 651-665. https://doi.org/10.1016/j.cam.2006.07.007
  15. G.He, Y.He, Z.Chen, A penalty finite volume method for the transient Navier-Stokes problem, Appl. Numer. Math., 58 (2008) 1583-1613. https://doi.org/10.1016/j.apnum.2007.09.006
  16. G.He, Y.He, X.Feng, Finite volume method based on stabilized finite elements for the nonstationary Navier-Stokes problem, Numer. Methods Partial Differential Eq., 23 (2007) 1167-1191. https://doi.org/10.1002/num.20216
  17. J.Heywood, R.Rannacher, Finite element approximation of the nonstationary Navier- Stokes problem I: regularity of solutions and second-order error estimates for spatial dis- cretization, SIAM J. Numer. Anal. 19 (1982) 275-311. https://doi.org/10.1137/0719018
  18. J.Li, Z.Chen, A new stabilized finite volume method for the stationary Stokes equations, Adv. Comput. Math., 30 (2009) 141-152. https://doi.org/10.1007/s10444-007-9060-5
  19. R.Li, Z.Chen, W.Wu, Generalized difference methods for differential equations, Marcel Dekker, New York, 2000.
  20. J.Li, Y.He, A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math. 214 (2008) 58-65. https://doi.org/10.1016/j.cam.2007.02.015
  21. H.Wu, R.Li, Error estimates for finite volume element methods for general second-order elliptic problems, Numer. Methods Partial Differential Eq., 19 (2003) 693-708. https://doi.org/10.1002/num.10068
  22. H.Notsu, M.Tabata, A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations, J. Sci. Comput., 38 (2009) 1-14. https://doi.org/10.1007/s10915-008-9217-5
  23. O.Pironneau, On the transport-diffusion algorithm and its applications to the Navier- Stokes equations, Numer. Math., 38 (1982) 309-332. https://doi.org/10.1007/BF01396435
  24. E.Suli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53 (1988) 459-483. https://doi.org/10.1007/BF01396329
  25. R. Temam, Navier-Stokes equation: Theory and numerical analysis (Third edition), North- Holland, Amsterdam, New York, Oxford, 1984.
  26. C.Wang, Characteristic finite analytic method (CFAM) for incompressible Navier-Stokes equations, ACTA Mech., 143 (2000) 57-66. https://doi.org/10.1007/BF01250017
  27. X.Ye, On the relationship between finite volume and finite element methods qpplied to the Stokes equations, Numer. Methods Partial Differential Eq., 5 (2001) 440-453.
  28. T.Zhang, Z.Y.Si, Y.N.He, A stabilized characteristic finite element method for the tran- sient Navier-Stokes equations, Int. J. Comput. Fluid Dyn., on line.